Condici\'on de Lorentz y ecuaciones de ondas electromagn\'eticas como propiedades emergentes del sistema de Maxwell

TL;DR

This paper explores how electromagnetic wave equations and the Lorentz condition emerge from Maxwell's system using quaternion analysis, linking solutions to quaternion equations and simplifying the system.

Contribution

It introduces a quaternion-based approach to derive wave equations and the Lorentz condition from Maxwell's equations, providing a novel mathematical framework.

Findings

Derived wave and Helmholtz equations from Maxwell's system.

Established connection between Maxwell solutions and quaternion equations.

Simplified Maxwell system using quaternion scalar and vector potentials.

Abstract

This article deals with the study of electromagnetic waves equations and the Lorentz condition, as emergent properties of Maxwell's system in the context of systems theory. To do this, the wave equations and the Helmholtz equation are first deduced. Using the displaced Dirac operator, which is closely related to the main vector calculation operators, it is possible to establish a direct connection between the solutions of the Maxwell time-harmonic system and two quaternion equations. Also, the application of the Lorentz condition to transform the time-harmonic Maxwell system into a simple quaternion equation based on the scalar and vector potentials is exposed.